Modeling and localized feedforward control of thermal deformations induced by a moving heat load

The method of images in thermoelasticity with an application to wafer heating

“…Note that it is often much easier to compute T pass ðx, tÞ and u pass ðx, tÞ resulting from Q pass ðx, tÞ than the solution u X, 1 ðx, tÞ resulting from Q X, 1 ðx, tÞ because u pass ðx, tÞ typically needs to be computed over a smaller time window and requires a mesh of a smaller part of the spatial domain than u X, 1 ðx, tÞ: This makes the construction of T X, 1 ðx, tÞ and u X, 1 ðx, tÞ through (19) and (22) attractive in many applications such as the wafer heating problem considered in Section 5.…”

“…In this application, a thin silicon disk, called the wafer, is exposed to a light source which creates a pattern of electronic connections (see Figure 1). Because the wafer is thin, a 2-D plane-stress model is used, just as in [11,19]. The temperature field T D ðx, tÞ and the displacement field u D ðx, tÞ resulting from the heat load Q D ðx, tÞ are thus solutions of (1) with BCs (6), where…”

“…When Q pass ðx, tÞ is the heat load applied to a single field centered at x ¼ 0 that is scanned in the positive x 2 -direction starting at time t ¼ 0, the i-th shift in space x i is the center of the i-th field, the i-th rotation matrix R i ¼ I when the i-th field is scanned in the positive x 2 -direction and R i ¼ ÀI when the i-th field is scanned in the negative x 2 -direction, and the i-th shift in time s i is the time at which the scanning of the i-th field starts. On R 2 , the temperature and displacement fields T D, 1 ðx, tÞ and u D, 1 ðx, tÞ resulting from the repetitive heat load Q D, 1 ðx, tÞ can thus be constructed from the responses T pass ðx, tÞ and u pass ðx, tÞ resulting from Q pass ðx, tÞ according to (19) and (22), respectively.…”

“…The first step to obtain T D, appr ðx, tÞ and u D, appr ðx, tÞ using the MoI is the construction of the temperature and displacement fields T D, 1 ðx, tÞ and u D, 1 ðx, tÞ on R 2 based on the single-field responses T pass ðx, tÞ and u pass ðx, tÞ according to (19) and (22). This requires the interpolation of T pass ðx, tÞ and u pass ðx, tÞ, which is done using MATLAB's griddedInterpolant function.…”

“…The heat load applied during each passing is the same up to a translation and/or rotation in space and a shift in time. Examples of such processes are the laser hardening of metals [12][13][14], additive manufacturing [15][16][17], and wafer heating [11,18,19]. In the latter application, a pattern of electronic connections is projected onto a silicon wafer, see Figure 1.…”